Here is a theorem in my lecture notes:
Let $S$ be a compatible $(d-1)$-dimensional $C^1$-hypersurface with $a, b \in C^1$ in a neighbourhood of $S$ and $g$ a $C^1$ function on $S$. For every $x_0 \in S$, there is a neighbourhood $\Omega$ of $x_0$ such that the Cauchy problem $$a(x, u) \cdot Du = b(x, u) \hspace{2mm} \text{in}\hspace{2mm} \Omega, \hspace{2mm} u = g \hspace{2mm} \text{on} \hspace{2mm} \Omega \cap S$$ has a unique solution $u \in C^1(\Omega)$
[Note: we define $S$ to be compatible with the equation if $\nu(x) \cdot a(x, g(x)) \neq 0 \forall x \in S$]
So I'm first trying to understand the statement here because it's quite abstract (to me).
So I thought to try some PDEs out and see if they match the conditions in the statement, but I'm struggling to apply the statement's conditions in practice.
For example, suppose I have the PDE $$\partial_{x_1} u + 2\partial_{x_2}u = u, u(z, 0) = h(z)$$ for $h \in C^1$.
In this case, what would be my $S$? Is it compatible, I'm not sure how to 'verify' the compatibility condition in general?
Are there any examples where the statement fail and why? (Well I know there are, but I'm not too sure why they contradict the statement).
If someone could provide some examples of first order PDEs for which $S$ is not compatible, that would be very helpful I think.